Recently, I saw this question How many graphs are there such that every vertex has degree at most 2? about finding the number of labeled and unlabeled graphs with vertices having degree at most two. Number of vertices is fixed to a given $n$.
My question is the following: Can this result be generalised to
- A) Determining the number of i) labeled, ii) unlabeled undirected graphs with $n$ vertices of degree at most 3
- B) Determining the number of i) labeled, ii) unlabeled undirected graphs with $n$ vertices of degree 2,3
Evaluating the generating function seems sufficient.
Apart from isolated points, I had some idea to determine A from B: For every vertex with degree 2 from B, we can add a line (which ends in some point), so the vertex is degree three. That way, we can construct all graphs with vertices including degree 1.